Concept explainers
The string shown in Figure P13.5 is driven at a frequency of 5.00 Hz. The amplitude of the motion is A = 12.0 cm, and the wave speed is v = 20.0 m/s. Furthermore, the wave is such that y = 0 at x = 0 and t = 0. Determine (a) the angular frequency and (b) the wave number for this wave. (c) Write an expression for the wave function. Calculate (d) the maximum transverse speed and (e) the maximum transverse acceleration of an element of the string.
Figure P13.5
(a)
The angular frequency of the wave.
Answer to Problem 5P
The angular frequency of the wave is
Explanation of Solution
Write the expression for the frequency of the string.
Here,
Solve equation (I) for
Conclusion:
Substitute
Therefore, the angular frequency of the wave is
(b)
The wave number of the wave.
Answer to Problem 5P
The wave number of the wave is
Explanation of Solution
Write the expression for the wavelength of the wave.
Here,
Solve equation (III) for
Write the expression for the wavelength in terms of speed of the wave.
Conclusion:
Substitute
Substitute
Therefore, The wave number of the wave is
(c)
Expression for the wave function.
Answer to Problem 5P
Expression for the wave function is
Explanation of Solution
The general form of a wave function can be represented as,
Here,
Conclusion:
Using initial conditions, to make this fit,
In this case, taking initial conditions, substitute
Therefore, Expression for the wave function is
(d)
The maximum transverse speed of the wave.
Answer to Problem 5P
The maximum transverse speed of the wave is
Explanation of Solution
Write the expression for the transverse speed.
Differentiate equation (VI) in equation (VII),
Conclusion:
The maximum magnitude is given by,
Substitute
Therefore, the maximum transverse speed of the wave is
(e)
The maximum transverse acceleration of an element of the string.
Answer to Problem 5P
The maximum transverse acceleration of an element of the string.is
Explanation of Solution
Write the expression for the transverse acceleration.
Use equation (VIII) in equation (VII),
Conclusion:
The maximum magnitude is given by,
Substitute
Therefore, The maximum transverse acceleration of an element of the string.is
Want to see more full solutions like this?
Chapter 13 Solutions
Bundle: Principles of Physics: A Calculus-Based Text, 5th + WebAssign Printed Access Card for Serway/Jewett's Principles of Physics: A Calculus-Based Text, 5th Edition, Multi-Term
- As in Figure P18.16, a simple harmonic oscillator is attached to a rope of linear mass density 5.4 102 kg/m, creating a standing transverse wave. There is a 3.6-kg block hanging from the other end of the rope over a pulley. The oscillator has an angular frequency of 43.2 rad/s and an amplitude of 24.6 cm. a. What is the distance between adjacent nodes? b. If the angular frequency of the oscillator doubles, what happens to the distance between adjacent nodes? c. If the mass of the block is doubled instead, what happens to the distance between adjacent nodes? d. If the amplitude of the oscillator is doubled, what happens to the distance between adjacent nodes? FIGURE P18.16arrow_forwardReview. A sphere of mass M is supported by a string that passes over a pulley at the end of a horizontal rod of length L (Fig. P14.25). The string makes an angle θ with the rod. The fundamental frequency of standing waves in the portion of the string above the rod is f. Find the mass of the portion of the string above the rod. Figure P14.25 Problems 25 and 26.arrow_forwardA string with a mass m = 8.00 g and a length L = 5.00 m has one end attached to a wall; the other end is draped over a small, fixed pulley a distance d = 4.00 m from the wall and attached to a hanging object with a mass M = 4.00 kg as in Figure P14.21. If the horizontal part of the string is plucked, what is the fundamental frequency of its vibration? Figure P14.21arrow_forward
- The string shown in Figure P16.11 is driven at a frequency of 5.00 Hz. The amplitude of the motion is A = 12.0 cm, and the wave speed is v = 20.0 m/s. Furthermore, the wave is such that y = 0 at x = 0 and t = 0. Determine (a) the angular frequency and (b) the wave number for this wave. (c) Write an expression for the wave function. Calculate (d) the maximum transverse speed and (e) the maximum transverse acceleration of an element of the string.arrow_forwardReview. A light string with a mass per unit length of 8.00 g/m has its ends tied to two walls separated by a distance equal to three-fourths the length of the string (Fig. P13.18). An object of mass m is suspended from the center of the string, putting a tension in the string. (a) Find an expression for the transverse wave speed in the string as a function of the mass of the hanging object. (b) What should be the mass of the object suspended from the string if the wave speed is to be 60.0 m/s? Figure P13.18arrow_forwardA block of mass m = 5.00 kg is suspended from a wire that passes over a pulley and is attached to a wall (Fig. P17.71). Traveling waves are observed to have a speed of 33.0 m/s on the wire. a. What is the mass per unit length of the wire? b. What would the speed of waves on the wire be if the suspended mass were decreased to 2.50 kg? FIGURE P17.71arrow_forward
- Review. A block of mass M, supported by a string, rests on a frictionless incline making an angle with the horizontal (Fig. P13.50). The length of the string is L, and its mass is m M. Derive an expression for the time interval required for a transverse wave to travel from one end of the string to the other. Figure P13.50arrow_forwardReview. A 12.0-kg object hangs in equilibrium from a string with a total length of L = 5.00 m and a linear mass density of = 0.001 00 kg/m. The string is wrapped around two light, frictionless pulleys that are separated by a distance of d = 2.00 m (Fig. P17.47a). (a) Determine the tension in the string. (b) At what frequency must the string between the pulleys vibrate to form the standing-wave pattern shown in Figure P 17.47b? Figure P17.47 Problem 47 and 48.arrow_forwardThe sinusoidal wave shown in Figure P13.41 is traveling in the positive x-direction and has a frequency of 18.0 Hz. Find the (a) amplitude, (b) wavelength, (c) period, and (d) speed of the wave. Figure P13.41arrow_forward
- Principles of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPhysics for Scientists and Engineers, Technology ...PhysicsISBN:9781305116399Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPhysics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage Learning
- College PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage LearningPhysics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPhysics for Scientists and Engineers with Modern ...PhysicsISBN:9781337553292Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning